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Notebook[{


Cell[CellGroupData[{Cell[TextData["Genetic Algorithms"], "Title",
  Evaluatable->False,
  AspectRatioFixed->True],

Cell[TextData[
"Written by Mats G. Bengtsson\nNational Defence Research Establishment\nBox \
1165, S-581 11 Linkoping\nSweden\nemail:  matben@lin.foa.se"], "Subsubtitle",
  Evaluatable->False,
  AspectRatioFixed->True],

Cell[TextData[
"This is intended to be a simple and pedagogical illustration of how genetic \
algorithms may be used for function optimization. The code is very slow so it \
is only intended for demos, not for production runs. For further info, see \
for instance the book by Goldberg.\n\nA genetic algorithm consists \
essentially of three parts:\n--   Selection; each string is selected with a \
probability proportional to its fitness value.\n--   Crossover; in a pair of \
selected strings, sition along the string is chosen, and the right and left \
part of each     string is swapped.\n--   Mutation; each gene is changed at \
random with a small probability."], "Text",
  Evaluatable->False,
  AspectRatioFixed->True],

Cell[CellGroupData[{Cell[TextData["Set up the problem"], "Subsection",
  Evaluatable->False,
  AspectRatioFixed->True],

Cell[TextData[
"In our case we have chosen the conventional binary coding. Each individual \
is coded as a binary string with length stringLength. The problem is limited \
to 1D, and for x from 0 to 1."], "Text",
  Evaluatable->False,
  AspectRatioFixed->True],

Cell[CellGroupData[{Cell[TextData[
"stringLength = 10;\nmutationRate = 0.002;\npopSize = 50;  \t\t(* must be a \
power of two *)"], "Input",
  InitializationCell->True,
  AspectRatioFixed->True],

Cell[TextData[
"General::spell1: \n   Possible spelling error: new symbol name \
\"stringLength\"\n     is similar to existing symbol \"StringLength\"."], 
  "Message",
  Evaluatable->False,
  AspectRatioFixed->True]}, Open]],

Cell[TextData[
"Create a random population. The population is defined on the real axis from \
0 to 1. The random values are approximated with binary strings given the \
stringLength."], "Text",
  Evaluatable->False,
  AspectRatioFixed->True],

Cell[TextData[
"popFloats = Table[ Random[], {popSize} ];\npopStrings = Map[ floatToBinary, \
popFloats ];"], "Input",
  InitializationCell->True,
  AspectRatioFixed->True],

Cell[TextData[
"Define the fitness function. It must be defined in the interval 0 to 1. The \
first one is a simple one, while the second function is a little bit more \
difficult."], "Text",
  Evaluatable->False,
  AspectRatioFixed->True],

Cell[TextData[
"Clear[fitnessFunction];\nfitnessFunction[x_] := Sin[ N[Pi] x ]"], "Input",
  InitializationCell->True,
  AspectRatioFixed->True],

Cell[TextData[
"Clear[fitnessFunction];\nfitnessFunction[x_] := Sin[N[Pi]x]*Mod[9x, 1]"], 
  "Input",
  InitializationCell->True,
  AspectRatioFixed->True]}, Open]],

Cell[CellGroupData[{Cell[TextData["Function definitions"], "Subsection",
  Evaluatable->False,
  AspectRatioFixed->True],

Cell[CellGroupData[{Cell[TextData["Simple crossover"], "Subsubsection",
  Evaluatable->False,
  AspectRatioFixed->True],

Cell[TextData[
"A random position is chosen along the strings. Create two new strings by \
swapping the parts before and after the cut respectively."], "Text",
  Evaluatable->False,
  AspectRatioFixed->True],

Cell[TextData[
"Clear[doSingleCrossover];\ndoSingleCrossover[ {string1_, string2_} ] := \
Module[{stle,cut,temp1,temp2},\n\tstle = Length[string1];\n\tcut = \
Random[Integer, {1,stle-1}];\n\ttemp1 = Join[ Take[string1, cut],\n\t\t\
Drop[string2, cut] ];\n\ttemp2 = Join[ Take[string2, cut],\n\t\tDrop[string1, \
cut] ];\n\t{temp1, temp2} ]"], "Input",
  InitializationCell->True,
  AspectRatioFixed->True]}, Open]],

Cell[CellGroupData[{Cell[TextData["Mutations"], "Subsubsection",
  Evaluatable->False,
  AspectRatioFixed->True],

Cell[TextData[
"Every gene may be excanged, from 0 to 1 and vice versa, with the probability \
mutationRate. The idea behind this is to introduce a mechanism for escaping \
from local minimas."], "Text",
  Evaluatable->False,
  AspectRatioFixed->True],

Cell[TextData[
"Clear[doMutation];\ndoMutation[string_] := Module[{tempstring,i},\n\t\
tempstring = string;\n\tDo[ If[ Random[] < mutationRate,\n\t\ttempstring[[i]] \
= 1 - tempstring[[i]] ],\n\t\t{i,stringLength} ];\n\ttempstring\n]"], "Input",\

  InitializationCell->True,
  AspectRatioFixed->True]}, Open]],

Cell[CellGroupData[{Cell[TextData["Selection"], "Subsubsection",
  Evaluatable->False,
  AspectRatioFixed->True],

Cell[TextData[
"Select two strings from the population, with a probability for each string \
proportional to its fitness value. Return only the two indices."], "Text",
  Evaluatable->False,
  AspectRatioFixed->True],

Cell[TextData[
"This computes the cumulative sum of the fitness values. It is necessary in \
the selection process."], "Text",
  Evaluatable->False,
  AspectRatioFixed->True],

Cell[TextData[
"Clear[doCumSumOfFitness];\ndoCumSumOfFitness := Module[{temp},\n\ttemp = \
0.0;\n\tTable[ temp += popFitness[[i]], {i, popSize} ]\n]"], "Input",
  InitializationCell->True,
  AspectRatioFixed->True],

Cell[TextData[
"Clear[doSingleSelection];\ndoSingleSelection := Module[{rfitness,ind},\n\t\
rfitness = Random[Real, {0, cumFitness[[popSize]]}];\n\tind = 1;\n\tWhile[ \
rfitness > cumFitness[[ind]], ind++ ];\n\tind--\n]"], "Input",
  InitializationCell->True,
  AspectRatioFixed->True],

Cell[TextData[
"This routine selects a pair of individuals with probability for each \
proportional to its fitness value."], "Text",
  Evaluatable->False,
  AspectRatioFixed->True],

Cell[TextData[
"Clear[selectPair];\nselectPair := Module[{ind1,ind2},\n\tind1 = \
doSingleSelection;\n\tWhile[ (ind2 = doSingleSelection) == ind1, ];\n\t{ind1, \
ind2}\n]"], "Input",
  InitializationCell->True,
  AspectRatioFixed->True],

Cell[TextData[
"This routine selects a pair at random with equal probability for each. This \
pair is removed from the population."], "Text",
  Evaluatable->False,
  AspectRatioFixed->True],

Cell[TextData[
"Clear[pickRandomPair];\npickRandomPair := Module[{ind1,ind2},\n\tind1 = \
Random[Integer, {1, popSize}];\n\tWhile[ (ind2 = Random[Integer, {1, \
popSize}]) == ind1, ];\n\t{ind1, ind2}\n]"], "Input",
  InitializationCell->True,
  AspectRatioFixed->True],

Cell[TextData[
"Swap strings. Update both the string and its corresponding fitness value."], 
  "Text",
  Evaluatable->False,
  AspectRatioFixed->True],

Cell[TextData[
"Clear[exchangeString];\nexchangeString[ind_, newstring_, newF_] := \
Module[{},\n\tpopStrings[[ind]] = newstring;\n\tpopFitness[[ind]] = newF;\n\
]"], "Input",
  InitializationCell->True,
  AspectRatioFixed->True]}, Open]],

Cell[CellGroupData[{Cell[TextData["Miscellaneous functions"], "Subsubsection",
  Evaluatable->False,
  AspectRatioFixed->True],

Cell[TextData[
"Clear[floatToBinary];\nfloatToBinary[x_] := Module[{temp},\n\ttemp = \
RealDigits[ x, 2 ];\n\tTake[ Join[ \n\t\tTable[0, {-temp[[2]]}], temp[[1]], \
Table[0, {stringLength}] ],\n\t\tstringLength ]\n] /; x<1"], "Input",
  InitializationCell->True,
  AspectRatioFixed->True],

Cell[TextData[
"Clear[binaryStringToFloat];\nbinaryStringToFloat[string_] :=\n\tN[Sum[ \
string[[i]]*2^(-i), {i, Length[string]} ]]"], "Input",
  InitializationCell->True,
  AspectRatioFixed->True],

Cell[TextData[
"Scale the fitness values linearly so that  max = 2 min. This is a good idea \
to do because otherwise the genetic material tends to be dominated by very \
few nonidentical members. There is always a balance between concentrating the \
search on the best strings, and having a search in a broad spectrum."], "Text",\

  Evaluatable->False,
  AspectRatioFixed->True],

Cell[TextData[
"Clear[renormalizeFitness];\nrenormalizeFitness[fitness_List] := \
Module[{minF,maxF,a,b},\n\tminF = Min[fitness];\n\tmaxF = Max[fitness];\n\ta \
= 0.5*maxF/(maxF+minF);\n\tb = (1-a)*maxF;\n\tMap[ a# + b &, fitness ]\n]"], 
  "Input",
  InitializationCell->True,
  AspectRatioFixed->True]}, Open]],

Cell[CellGroupData[{Cell[TextData["Initialize"], "Subsubsection",
  Evaluatable->False,
  AspectRatioFixed->True],

Cell[TextData[
"This initializes everything except for the population itself."], "Text",
  Evaluatable->False,
  AspectRatioFixed->True],

Cell[TextData[
"Clear[doInitialize];\ndoInitialize := Module[{i},\n\tpopFitness = Table[\n\t\
\tfitnessFunction[ binaryStringToFloat[ popStrings[[i]] ] ],\n\t\t{i,popSize} \
];\n\tcumFitness = doCumSumOfFitness;\n\tlistOfCumFitness = \
{cumFitness[[popSize]]};\n\thistoryOfPop = { popStrings };\n]"], "Input",
  InitializationCell->True,
  AspectRatioFixed->True]}, Open]]}, Open]],

Cell[CellGroupData[{Cell[TextData["Main"], "Subsection",
  Evaluatable->False,
  AspectRatioFixed->True],

Cell[CellGroupData[{Cell[TextData["Make a new generation; update asynchronously"], "Subsubsection",
  Evaluatable->False,
  AspectRatioFixed->True],

Cell[TextData[
"There are actually two ways you can do it. The asynchronous mode corresponds \
to picking the parents, do crossover and mutation, and exchange the children \
with a randomly picked pair. This proceeds pair by pair."], "Text",
  Evaluatable->False,
  AspectRatioFixed->True],

Cell[CellGroupData[{Cell[TextData[
"Clear[updateGenerationAsync];\n\nupdateGenerationAsync := \
Module[{parents,children,childrenF,rm,i},\n\tDo[\tparents = selectPair;\n\t\t\
children = doSingleCrossover[ {popStrings[[ parents[[1]] ]], \n\t\t\t\
popStrings[[ parents[[2]] ]]} ];\n\t\tchildren = Map[ doMutation, children ]; \
\n\t\tchildrenF = fitnessFunction[ \n\t\t\tMap[ binaryStringToFloat, children \
] ];\n\t\trm = pickRandomPair;\n\t\tDo[exchangeString[ rm[[i]], \
children[[i]], childrenF[[i]] ],\n\t\t\t{i,2} ];\n\t\tcumFitness = \
doCumSumOfFitness;\n\t\tAppendTo[ listOfCumFitness, cumFitness[[popSize]] ],\n\
\t{popSize/2} ];\n];"], "Input",
  InitializationCell->True,
  AspectRatioFixed->True],

Cell[TextData[
"General::spell1: \n   Possible spelling error: new symbol name \"childrenF\"\
\n     is similar to existing symbol \"children\"."], "Message",
  Evaluatable->False,
  AspectRatioFixed->True]}, Open]]}, Open]],

Cell[CellGroupData[{Cell[TextData["Make a new generation; update synchronously"], "Subsubsection",
  Evaluatable->False,
  AspectRatioFixed->True],

Cell[TextData[
"This represents the other way to do it. The complete parent population is \
chosen in a single sweep, and the children constitutes the new population."], 
  "Text",
  Evaluatable->False,
  AspectRatioFixed->True],

Cell[TextData[
"Clear[updateGenerationSync];\n\nupdateGenerationSync := \
Module[{parentsid,children,ip},\n\tparentsid = {};\n\tDo[\tAppendTo[ \
parentsid, selectPair ], {popSize/2} ];\n\tchildren = {};\n\tDo[ AppendTo[ \
children, \n\t\t\tdoSingleCrossover[ {popStrings[[parentsid[[ip,1]]]], \n\t\t\
\tpopStrings[[parentsid[[ip,2]]]]} ] ],\n\t\t{ip, popSize/2} ];\n\tpopStrings \
= Flatten[ children, 1];\n\tpopStrings = Map[ doMutation, popStrings ]; \n\t\
popFitness = fitnessFunction[ \n\t\tMap[ binaryStringToFloat, popStrings ] ];\
\n\tpopFitness = renormalizeFitness[ popFitness ];\n\tcumFitness = \
doCumSumOfFitness;\t\n];"], "Input",
  InitializationCell->True,
  AspectRatioFixed->True]}, Open]],

Cell[CellGroupData[{Cell[TextData["This is Main"], "Subsubsection",
  Evaluatable->False,
  AspectRatioFixed->True],

Cell[TextData["doInitialize;"], "Input",
  AspectRatioFixed->True],

Cell[TextData[
"Do[\tupdateGenerationSync;\n\tAppendTo[ historyOfPop, popStrings ];\n\t\
AppendTo[ listOfCumFitness, cumFitness[[popSize]] ],\n\t{10} ];"], "Input",
  AspectRatioFixed->True]}, Open]],

Cell[CellGroupData[{Cell[TextData["Plot results"], "Subsubsection",
  Evaluatable->False,
  AspectRatioFixed->True],

Cell[TextData[
"This is the sum of fitness for the whole population plotted versus \
generation number."], "Text",
  Evaluatable->False,
  AspectRatioFixed->True],

Cell[CellGroupData[{Cell[TextData[
"ListPlot[ listOfCumFitness, PlotJoined->True, PlotRange->All ];"], "Input",
  AspectRatioFixed->True],

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